More Inequalities

Finishing up probability theory a day late. Here are some named inequalities that I'll have to memorize:

Holder's inequality: |E(XY)| ≤ E(|XY|) ≤ (E(|X|^1/p))^p(E(|X|^1/(1-p)))^1-p, p∈(0,1).

when p = 1/2, this is the Cauchy-Schwarz inequality, which I've already written about. Centering X and Y about their respective means and leaving p = 1/2 yields:

Covariance inequality: (Cov(X, Y))² ≤ σ_X²σ_Y²

Another sometimes useful case which can be fairly easily derived from Holder is:

Liapounov's inequality: (E|X|^r)^1/r ≤ (E|X|^s)^1/s, 1 < r < s

Similar looking, but proven differently is:

Minkowski's inequality: (E|X+Y|^p)^1/p ≤ (E|X|^p)^1/p + (E|Y|^p)^1/p.

Jensen's inequality: Eg(X) > g(EX) for any convex function g.